3.446 \(\int \frac{\sqrt{\sin ^{-1}(a x)}}{(c-a^2 c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=185 \[ -\frac{a \sqrt{1-a^2 x^2} \text{Unintegrable}\left (\frac{x}{\left (1-a^2 x^2\right )^2 \sqrt{\sin ^{-1}(a x)}},x\right )}{6 c^2 \sqrt{c-a^2 c x^2}}-\frac{a \sqrt{1-a^2 x^2} \text{Unintegrable}\left (\frac{x}{\left (1-a^2 x^2\right ) \sqrt{\sin ^{-1}(a x)}},x\right )}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \sqrt{\sin ^{-1}(a x)}}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{x \sqrt{\sin ^{-1}(a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]

[Out]

(x*Sqrt[ArcSin[a*x]])/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x*Sqrt[ArcSin[a*x]])/(3*c^2*Sqrt[c - a^2*c*x^2]) - (a*S
qrt[1 - a^2*x^2]*Unintegrable[x/((1 - a^2*x^2)^2*Sqrt[ArcSin[a*x]]), x])/(6*c^2*Sqrt[c - a^2*c*x^2]) - (a*Sqrt
[1 - a^2*x^2]*Unintegrable[x/((1 - a^2*x^2)*Sqrt[ArcSin[a*x]]), x])/(3*c^2*Sqrt[c - a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.188572, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt{\sin ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[ArcSin[a*x]]/(c - a^2*c*x^2)^(5/2),x]

[Out]

(x*Sqrt[ArcSin[a*x]])/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x*Sqrt[ArcSin[a*x]])/(3*c^2*Sqrt[c - a^2*c*x^2]) - (a*S
qrt[1 - a^2*x^2]*Defer[Int][x/((1 - a^2*x^2)^2*Sqrt[ArcSin[a*x]]), x])/(6*c^2*Sqrt[c - a^2*c*x^2]) - (a*Sqrt[1
- a^2*x^2]*Defer[Int][x/((1 - a^2*x^2)*Sqrt[ArcSin[a*x]]), x])/(3*c^2*Sqrt[c - a^2*c*x^2])

Rubi steps

\begin{align*} \int \frac{\sqrt{\sin ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{x \sqrt{\sin ^{-1}(a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 \int \frac{\sqrt{\sin ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac{\left (a \sqrt{1-a^2 x^2}\right ) \int \frac{x}{\left (1-a^2 x^2\right )^2 \sqrt{\sin ^{-1}(a x)}} \, dx}{6 c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sqrt{\sin ^{-1}(a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 x \sqrt{\sin ^{-1}(a x)}}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{\left (a \sqrt{1-a^2 x^2}\right ) \int \frac{x}{\left (1-a^2 x^2\right )^2 \sqrt{\sin ^{-1}(a x)}} \, dx}{6 c^2 \sqrt{c-a^2 c x^2}}-\frac{\left (a \sqrt{1-a^2 x^2}\right ) \int \frac{x}{\left (1-a^2 x^2\right ) \sqrt{\sin ^{-1}(a x)}} \, dx}{3 c^2 \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 1.721, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\sin ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[ArcSin[a*x]]/(c - a^2*c*x^2)^(5/2),x]

[Out]

Integrate[Sqrt[ArcSin[a*x]]/(c - a^2*c*x^2)^(5/2), x]

________________________________________________________________________________________

Maple [A]  time = 0.296, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{\arcsin \left ( ax \right ) } \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)^(1/2)/(-a^2*c*x^2+c)^(5/2),x)

[Out]

int(arcsin(a*x)^(1/2)/(-a^2*c*x^2+c)^(5/2),x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^(1/2)/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^(1/2)/(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(a*x)**(1/2)/(-a**2*c*x**2+c)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\arcsin \left (a x\right )}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^(1/2)/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(arcsin(a*x))/(-a^2*c*x^2 + c)^(5/2), x)